Question

Solve this:

8. A sonometer wire of length 1.5 m is made of steel. The tension in it produces an elastic strain of 1%. What is the fundamental frequency of steel, if density and elasticity of steel are 7.7 $\times$ 10³ kg/m³ and 2.2 $\times$ 10¹¹ N/m² respectively?

(a) 188.5 Hz (b) 178.2 Hz
(c) 200.5 Hz (d) 770 Hz​

Answer 1

Dear Student ,
Transverse waves on a Sonometer wire/bridge.

Young's modulus Y = Stress / Strain = 2.2 x 10¹¹ N/m²
Stress = Tension / Cross section area = T / A
Strain = 1 % = 0.01
2.2 x 10¹¹ = (T / A ) / 0.01 = T / (0.01 A)

T = 2.2 x 10⁹ x A Newtons
L = 1.5 meter

Volume density d = 7.7 x 10³ kg/m³
Linear density μ = volume density x Area = d x A = 7.7 x 10³ x A kg /meter

velocity of transverse wave on the Sonometer wire = √(T/μ)
v = √(2.2x10⁹ / 7.7x10³) m/sec
= 534.52 m/sec

Fundamental note has a wavelength = λ = 2 x L = 3 meters. As the given wire vibrates with one loop. Nodes at both ends and anti-node at the centre. So half of the wavelength is equal to the length of the wire.

So frequency = v / λ = 178.17 Hz

Regards